1 INTRODUCTION TO LASER OPERATION
1.5 EINSTEIN THEORY OF LIGHT – MATTER INTERACTIONS
The atoms of a laser undergo repeated quantum jumps and so act as microscopic transducers. That is, each atom accepts energy and jumps to a higher orbit as a result of some input or “pumping” process and converts it into other forms of energy—for example, into light energy (photons)—when it jumps to a lower orbit. At the same time, each atom must deal with the photons that have been emitted earlier and reflected back by the mirrors. These prior photons, already channeled along the cavity axis, are the origin of the stimulated component to the atom’s emission of subsequent photons.
In Fig. 1.12 we indicate some ways in which energy conversion can occur. For sim- plicity we focus our attention on quantum jumps between two energy levels, 1 and 2, of an atom. The five distinct energy conversion diagrams of Fig. 1.12 are interpreted as follows:
(a) Absorption of an incrementDE¼E22E1of energy from the pump: The atom is raised from level 1 to level 2. In other words, an electron in the atom jumps from an inner orbit to an outer orbit.
(b) Spontaneous emission of a photon of energy hn¼E22E1: The atom jumps down from level 2 to the lower level 1. The process occurs “spontaneously” with- out any external influence.
(c) Stimulated emission: The atom jumps down from energy level 2 to the lower level 1, and the emitted photon of energy hn¼E22E1 is an exact replica of a photon already present. The process is induced, or stimulated, by the incident photon.
(d) Absorption of a photon of energyhn¼E22E1: The atom jumps up from level 1 to the higher level 2. As in (c), the process is induced by an incident photon.
(e) Nonradiative deexcitation: The atom jumps down from level 2 to the lower level 1, but no photon is emitted so the energyE22E1must appear in some other form [e.g., increased vibrational or rotational energy in the case of a molecule, or rearrangement (“shakeup”) of other electrons in the atom].
All these processes occur in the gain medium of a laser. Lasers are often classified according to the nature of the pumping process (a) which is the source of energy for the output laser beam. In electric-discharge lasers, for instance, the pumping occurs
1.5 EINSTEIN THEORY OF LIGHT – MATTER INTERACTIONS 11
as a result of collisions of electrons in a gaseous discharge with the atoms (or molecules) of the gain medium. In an optically pumped laser the pumping process is the same as the absorption process (d), except that the pumping photons are supplied by a lamp or perhaps another laser. In a diode laser an electric current at the junction of two different semiconductors produces electrons in excited energy states from which they can jump into lower energy states and emit photons.
This quantum picture is consistent with a highly simplified description of laser action.
Suppose that lasing occurs on the transition defined by levels 1 and 2 of Fig. 1.12. In the most favorable situation the lower level (level 1) of the laser transition is empty. To maintain this situation a mechanism must exist to remove downward jumping electrons from level 1 to another level, say level 0. In this situation there can be no detrimental absorption of laser photons due to transitions upward from level 1 to level 2. In practice the number of electrons in level 1 cannot be exactly zero, but we will assume for sim- plicity that the rate of deexcitation of the lower level 1 is so large that the number of atoms remaining in that level is negligible compared to the number in level 2; this is a reasonably good approximation for many lasers. Under this approximation laser action can be described in terms of two “populations”: the numbernof atoms in the upper level 2 and the numberqof photons in the laser cavity.
The number of laser photons in the cavity changes for two main reasons:
(i) Laser photons are continually being added because of stimulated emission.
(ii) Laser photons are continually being lost because of mirror transmission, scattering or absorption at the mirrors, etc.
E2
E1
E2
E1
hn
E2 hn
hn hn
(a) (b)
(c)
(d) (e)
E1 hn
E2
E1
E2
E1
Figure 1.12 Energy conversion processes in a lasing atom or molecule: (a) absorption of energy DE¼E22E1from the pump; (b) spontaneous emission of a photon of energyDE; (c) stimulated emis- sion of a photon of energyDE; (d) absorption of a photon of energyDE; (e) nonradiative deexcitation.
Thus, we can write a (provisional) equation for the rate of change of the number of photons, incorporating the gain and loss described in (i) and (ii) as follows:
dq
dt ¼anqbq: (1:5:1)
That is, the rate at which the number of laser photons changes is the sum of two separate rates: the rate of increase (amplification or gain) due to stimulated emission, and the rate of decrease (loss) due to imperfect mirror reflectivity.
As Eq. (1.5.1) indicates, the gain of laser photons due to stimulated emission is not only proportional to the numbernof atoms in level 2, but also to the numberqof photons already in the cavity. The efficiency of the stimulated emission process depends on the type of atom used and other factors. These factors are reflected in the size of the ampli- fication orgain coefficient a. The rate of loss of laser photons is simply proportional to the number of laser photons present.
We can also write a provisional equation forn. Both stimulated and spontaneous emission causento decrease (in the former case in proportion toq, in the latter case not), and the pump causesnto increase at some rate we denote byp. Thus, we write
dn
dt ¼ anqfnþp: (1:5:2) Note that the first term appears in both equations, but with opposite signs. This reflects the central role of stimulated emission and shows that the decrease ofn(excited atoms) due to stimulated emission corresponds precisely to the increase ofq(photons).
Equations (1.5.1) and (1.5.2) describe laser action. They show how the numbers of lasing atoms and laser photons in the cavity are related to each other. They do not indi- cate what happens to the photons that leave the cavity, or what happens to the atoms when their electrons jump to some other level. Above all, they do not tell how to evaluate the coefficientsa,b,f,p. They must be taken only as provisional equations, not well justified although intuitively reasonable.
It is important to note that neither Eq. (1.5.1) nor (1.5.2) can be solved independently of the other. That is, (1.5.1) and (1.5.2) arecoupledequations. The coupling is due phys- ically to stimulated emission: The lasing atoms of the gain medium can increase the number of photons via stimulated emission, but by the same process the presence of photons will also decrease the number of atoms in the upper laser level. This coupling between the atoms and the cavity photons is indicated schematically in Fig. 1.13.
We also note that Eqs. (1.5.1) and (1.5.2) are nonlinear. The nonlinearity (the product of the two variables nq) occurs in both equations and is another manifestation of
Atoms
Cavity photons Equation
(1.5.1)
Equation (1.5.2)
Figure 1.13 Self-consistent pair of laser equations.
1.5 EINSTEIN THEORY OF LIGHT – MATTER INTERACTIONS 13
stimulated emission. No established systematic methods exist for solving nonlinear differential equations, and there is no known general solution to these laser equations.
However, they have a number of well-defined limiting cases of some practical importance, and some of these do have known solutions. The most important case is steady state.
In steady state we can put bothdq/dtanddn/dtequal to zero. Then (1.5.1) reduces to n¼b
a;nt, (1:5:3)
which can be recognized as athresholdrequirement on the number of upper-level atoms.
That is, ifn,b/a, thendq/dt,0, and the number of photons in the cavity decreases, terminating laser action. The steady state of (1.5.2) also has a direct interpretation.
Fromdn/dt¼0 andn¼nt¼b/awe find q¼ p
b f
a: (1:5:4)
This equation establishes a threshold for the pumping rate, since the number of photons q cannot be negative. Thus, the minimum or threshold value of p compatible with steady-state operation is found by puttingq¼0:
pt ¼ f b
a ¼f nt: (1:5:5)
In words, the threshold pumping rate just equals the loss rate per atom times the number of atoms present at threshold.
In Chapters 4 – 6 we will return to a discussion of laser equations. We will deal there with steady state as well as many other aspects of laser oscillation in two-level, three- level, and four-level quantum systems.